Nonlinear delay reaction–diffusion equations: Traveling-wave solutions in elementary functions

The paper deals with nonlinear delay reaction–diffusion equations of the form ut=auxx+F(u,ū), where u=u(x,t)u=u(x,t) and ū=u(x,t−τ), with ττ denoting the delay time. We present a number of traveling-wave solutions of the form u=w(z)u=w(z), z=kx+λtz=kx+λt, that can be represented in terms of elementary functions. We consider equations with quadratic, power-law, exponential and logarithmic nonlinearities as well as more complex equations with the kinetic function dependent on one to four arbitrary functions of a single argument. All of the solutions obtained involve free parameters and so may be suitable for solving certain model problems as well as testing numerical and approximate analytical methods for delay reaction–diffusion equations and more complex nonlinear delay PDEs.